seismic behavior and design of boundary frame members of

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Seismic Behavior and Design of Boundary Frame Members of Steel Plate Shear Walls
by Bing Qu and Michel Bruneau
Technical Report MCEER-08-0012
April 26, 2008
This research was conducted at the University at Buffalo, State University of New York and was supported primarily by the Earthquake
Engineering Research Centers Program of the National Science Foundation under award number EEC 9701471.
NOTICE This report was prepared by the University at Buffalo, State University of New York as a result of research sponsored by MCEER through a grant from the Earthquake Engineering Research Centers Program of the National Sci- ence Foundation under NSF award number EEC-9701471 and other sponsors. Neither MCEER, associates of MCEER, its sponsors, the University at Buffalo, State University of New York, nor any person acting on their behalf:
a. makes any warranty, express or implied, with respect to the use of any information, apparatus, method, or process disclosed in this report or that such use may not infringe upon privately owned rights; or
b. assumes any liabilities of whatsoever kind with respect to the use of, or the damage resulting from the use of, any information, apparatus, method, or process disclosed in this report.
Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect the views of MCEER, the National Science Foundation, or other sponsors.
Seismic Behavior and Design of Boundary Frame Members of Steel Plate Shear Walls
by
Publication Date: April 26, 2008 Submittal Date: March 1, 2008
Technical Report MCEER-08-0012
Task Number 10.2.1
1 Assistant Professor, Department of Civil and Environmental Engineering, California Polytechnic State University; Former Graduate Student, Department of Civil, Struc- tural, and Environmental Engineering, University at Buffalo, State University of New York
2 Professor, Department of Civil, Structural, and Environmental Engineering, Univer- sity at Buffalo, State University of New York
MCEER University at Buffalo, State University of New York Red Jacket Quadrangle, Buffalo, NY 14261 Phone: (716) 645-3391; Fax (716) 645-3399 E-mail: [email protected]; WWW Site: http://mceer.buffalo.edu
NTIS DISCLAIMER
! This document has been reproduced from the best copy furnished by the sponsoring agency.
iii
Preface
The Multidisciplinary Center for Earthquake Engineering Research (MCEER) is a national center of excellence in advanced technology applications that is dedicated to the reduction of earthquake losses nationwide. Headquartered at the University at Buffalo, State University of New York, the Center was originally established by the National Science Foundation in 1986, as the National Center for Earthquake Engineering Research (NCEER).
Comprising a consortium of researchers from numerous disciplines and institutions throughout the United States, the Center’s mission is to reduce earthquake losses through research and the application of advanced technologies that improve engineering, pre- earthquake planning and post-earthquake recovery strategies. Toward this end, the Cen- ter coordinates a nationwide program of multidisciplinary team research, education and outreach activities.
MCEER’s research is conducted under the sponsorship of two major federal agencies: the National Science Foundation (NSF) and the Federal Highway Administration (FHWA), and the State of New York. Signifi cant support is derived from the Federal Emergency Management Agency (FEMA), other state governments, academic institutions, foreign governments and private industry.
-
Intelligent Response and Recovery
iv
A cross-program activity focuses on the establishment of an effective experimental and analytical network to facilitate the exchange of information between researchers located in various institutions across the country. These are complemented by, and integrated with, other MCEER activities in education, outreach, technology transfer, and industry partnerships.
This report presents the results of an analytical study to investigate the behavior of horizontal and vertical boundary frame members that may impact the performance of Steel Plate Shear Walls (SPSWs). New analytical models were developed for horizontal boundary frame members to calculate the plastic moment and resulting strength reduction caused by biaxial internal stress conditions, and to revisit and develop improved capacity design procedures that account for these reduced plastic moments. The models incorporate observations made in a companion experimental study (see Technical Report MCEER-08-0010). Next, the adequacy of a fl exibility limit for the design of vertical boundary frame members specifi ed in current design codes was assessed using the new models. The contribution of the boundary frame moment resisting action and infi ll panel tension fi eld action to the overall plastic strength of SPSWs was investigated.
v
ABSTRACT
A Steel Plate Shear Wall (SPSW) consists of infill steel panels surrounded by columns, called
Vertical Boundary Elements (VBEs), and beams, called Horizontal Boundary Elements
(HBEs). Those infill panels are allowed to buckle in shear and subsequently form diagonal
tension field actions to resist the lateral loads applied on the structure. Research conducted
since the early 1980s has shown that this type of system can exhibit high initial stiffness,
behave in a ductile manner, and dissipate significant amounts of hysteretic energy, which
make it a suitable option for the design of new buildings as well as for the retrofit of existing
constructions. However, some obstacles still exist impeding more widespread acceptance of
this system. For example, there remain uncertainties regarding the seismic behavior and
design of boundary frame members of SPSWs. This report presents analytical work
conducted to investigate the behavior and design of boundary frame members of SPSWs.
First, analytical models were developed to calculate the HBE plastic moment accounting for
the reduction in strength due to the presence of biaxial internal stress conditions followed by
development of analytical models, which take into account the reduced plastic moments of
HBEs, to estimate the design forces for intermediate HBEs to reliably achieve capacity design.
Those models combine the assumed plastic mechanism with a linear beam model of
intermediate HBE considering fully yielded infill panels, and are able to prevent in-span
plastic hinges. The above advances with regard to HBE behavior make it possible to
investigate and explain the observed intermediate HBE failure.
In addition, the work presented in this report assesses the adequacy of a flexibility limit for
VBE design specified by the current design codes using new analytical models developed to
prevent the undesirable in-plane and out-of-plane performances of VBEs.
Furthermore, this report investigates the relative and respective contributions of boundary
frame moment resisting action and infill panel tension field action to the overall plastic
strength of SPSWs, followed by a proposed procedure to make use of the strength provided
by the boundary frame moment resisting action. Future work needed to provide greater
insight on SPSW designs is also identified.
vii
ACKNOWLEDGEMENTS
This work was financially supported in whole by the Earthquake Engineering Research Centers
Program of the National Science Foundation under Award Number ECC-9701471 to the
Multidisciplinary Center for Earthquake Engineering Research. However, any opinions, findings,
conclusions, and recommendations presented in this document are those of the writers and do not
necessarily reflect the views of the sponsors.
ix
2 PAST RESEARCH ON STEEL PLATE SHEAR WALLS 5
2.1 Introduction 5
2.3 Timler and Kulak (1983) 7
2.4 Driver, Kulak, Kennedy and Elwi (1997) 7
2.5 Berman and Bruneau (2003) 9
2.6 Behbahanifard, Grondin and Elwi (2003) 11
2.7 Kharrazi, Ventura, Prion, and Sabouri-Ghomi (2004) 12
2.8 Vian and Bruneau (2005) 13
2.9 Lopez-Garcia and Bruneau (2006) 14
2.10 Shishkin, Driver and Grondin (2005) 15
2.11 Purba and Bruneau (2006) 16
2.12 Berman and Bruneau (2008) 17
3 PLASTIC MOMENT OF HORIZONTAL BOUNDARY ELEMENTS 19
3.1 Introduction 19
3.2 Loading Characteristics of HBE Cross-Sections 20
3.3 Plastic Moment of Wide Flange Members in Moment Frame 22
3.4 Reduced Yield Strength in HBE Web 24
3.5 Intermediate HBE under Equal Top and Bottom Tension Fields 30
3.5.1 Derivation of Plastic Moment 30
3.5.2 FE Verification 33
3.5.3 Effects of Shear Direction 36
3.6 Intermediate HBE under Unequal Top and Bottom Tension Fields 36
3.6.1 Derivation of Plastic Moment under Positive Flexure 37
3.6.2 Derivation of Plastic Moment under Negative Flexure 40
3.6.3 FE Verification 41
3.7 Additional Discussions on Anchor HBEs 51
3.8 Summary 57
4.1 Introduction 59
4.2 Expected Mechanism of SPSW and Infill Panel Yield Force 60
4.3 Axial Force in Intermediate HBE 62
4.3.1 Axial Effects Due to Boundary Moment Frame Sway 63
4.3.2 Axial Effects of Horizontal Tension Field Components on VBEs 64
4.3.3 Axial Effects of Vertical Tension Field Components on HBE 64
4.3.4 Axial Effects of Horizontal Tension Field Components on HBE 66
4.3.5 Resulting Axial Force in HBE 69
4.4 Shear Force in Intermediate HBE 70
4.4.1 Shear Effects Only Due to Infill Panel Yield Forces 71
4.4.1.1 Superposing Shear Effects from Sub-Tension Fields 71
4.4.1.2 Combining Shear Effects from Tension Field Components 79
4.4.1.3 FE Verification of Shear Effects due to Tension Fields 81
4.4.2 Shear Effects Due to Boundary Frame Sway Action Alone 83
4.4.3 Resulting Shear Force in HBE 84
4.5 Prevention of In-Span HBE Plastic Hinge 86
4.5.1 Moment Diagram of Intermediate HBE 86
4.5.2 Procedure to Avoid In-Span Plastic Hinge 88
xi
4.8 Capacity Design Procedure for Intermediate HBEs 101
4.9 Examination of Intermediate HBE Fractures in Tests 106
4.10 Summary 107
5.1 Introduction 109
5.2 Review of Flexibility Factor in Plate Girder Theory 110
5.3 Flexibility Limit for VBE Design 116
5.4 Prevention of VBE In-Plane Shear Yielding 119
5.4.1 Shear Demand and Strength of VBE 119
5.4.2 Observation of VBE Shear Yielding in Past Testing 121
5.5 VBE Out-of-Plane Buckling Strength 128
5.5.1 Analytical Models for Out-of-plane Buckling Strength of VBEs 128
5.5.1.1 Free Body Diagrams of VBEs 128
5.5.1.2 Energy Method and Boundary Conditions 130
5.5.1.3 Out-of-Plane Buckling Strength of VBE - Case A 131
5.5.1.4 Out-of-Plane Buckling Strength of VBE - Case B 137
5.5.1.5 Out-of-Plane Buckling Strength of VBE - Case C 139
5.5.1.6 Out-of-Plane Buckling Strength of VBE - Case D 142
5.5.2 Review of Out-of-Plane Buckling of VBEs in Past Tests 145
5.6 Summary 151
BOUNDARY FRAME MOMENT RESISTING ACTION 153
6.1 Introduction 153
xii
6.4 SPSW Overstrength and Balanced Design 158
6.4.1 Single-Story SPSW 158
6.4.2 Multistory SPSW 166
6.5 Boundary Frame Design of SPSWs Having Weak Infill Panels 171
6.5.1 Design Method I 173
6.5.2 Design Method II 173
6.5.3 Design Method III 174
6.6 Case Study 175
6.6.2 Analytical Model and Artificial Ground Motions 178
6.6.3 Result Comparison 180
6.7 Further Consideration 182
FOR FUTURE RESEARCH 185
8 REFERENCES 189
Appendix A SHEAR EFFECTS AT THE END OF INTERMEDIATE HBE
DUE TO TENSION FIELDS 193
Appendix B 197
Appendix C 209
2-1 Schematic of Strip Model (Thorburn et al. 1983) 6
2-2 FE Model of Test Specimen (Driver et al. 1997) 8
2-3 Strip Model of Driver et al. Specimen (Driver et al. 1997) 9
2-4 Single Story SPSW Collapse Mechanism (Berman and Bruneau 2003) 10
2-5 Example of Plastic Collapse Mechanism for Multi-story SPSWs
(Berman and Bruneau 2003) 10
2-6 FE Model of Test Specimen (Behbahanifard et al. 2003) 11
2-7 Free Body Diagram of Anchor HBE 13
2-8 Schematic of the Modified Strip Model (Shishkin et al. 2005) 16
2-9 VBE Free Body Diagrams (Berman and Bruneau 2008) 18
3-1 Typical Multistory SPSW 20
3-2 Shear Stress and Loading at HBE Ends 21
3-3 Example of Plastic Resistance of a Wide Flange Structural Shape
Subjected to Flexure, Axial, and Shear Forces 24
3-4 Loading of Intermediate HBE Segment and
Mohr's Circles of the Elements on the Web 25
3-5 Reduced Yield Strength per the Von Mises Criterion in Plane Stress 27
3-6 Axial Yield Strength under Combinations of Shear and
Vertical Stresses per the Von Mises Yield Criterion 29
3-7 Stress Diagrams of Intermediate HBE Cross-Section under Flexure,
Axial Compression, Shear Force, and Vertical Stresses
due to Equal Top and Bottom Tension Fields 31
3-8 FE Model of Intermediate HBE Segment under Flexure,
due to Equal Top and Bottom Tension Fields 33
xiv
Analytical Predictions versus FE Results 35
3-10 Effects of Shear Direction on Cross-Section Plastic Moment
Reduction Factor of Intermediate HBE 36
3-11 Stress Diagrams of Intermediate HBE Cross-Section under
Positive Flexure, Axial Compression, Shear Force,
and Vertical Stresses due to Unequal Top and Bottom Tension Fields 37
3-12 Stress Diagrams of Intermediate HBE Cross-Section
under Negative Flexure, Axial Compression, Shear Force,
and Vertical Stresses due to Unequal Top and Bottom Tension Fields 40
3-13 FE Model of Intermediate HBE under Positive Flexure,
due to Unequal Top and Bottom Tension Fields 42
3-14 FE Model of Intermediate HBE under Negative Flexure,
due to Unequal Top and Bottom Tension Fields 42
3-15 Plastic Moment Reduction Factor of Intermediate HBE Cross-Section
under Positive Flexure, Axial Compression, Shear Force, and
Linear Vertical Stresses: Analytical Predictions versus FE Results 45
under Negative Flexure, Axial Compression, Shear Force,
and Linear Vertical Stress: Analytical Predictions versus FE Results 46
3-17 Simplification of Vertical Stress Distribution 48
under Positive Flexure, Axial Compression, Shear Force, and Linear
Vertical Stresses: Analytical Predictions versus Simplified Approach 49
xv
3-20 Plastic Moment Reduction Factor of Anchor HBE Cross-Section under
Positive Flexure, Axial Compression, Shear Force, and
Negative Flexure, Axial Compression, Shear Force, and
Positive Flexure, Axial Compression, Shear Force, and Linear
3-23 Plastic Moment Reduction Factor of Anchor HBE Cross-Section
under Negative Flexure, Axial Compression, Shear Force, and Linear
4-1 Uniform Yielding Mechanism of a Multistory SPSW
(adapted from Berman and Bruneau 2003) 60
4-2 Decomposition of SPSW Free Body Diagrams:
(a) Typical SPSW;
of Infill Panel Yield Forces on VBEs;
(e) Boundary Frame with Vertical Components
of Infill Panel Yield Forces on HBEs;
(f) Boundary Frame with Horizontal and Vertical Components
of Infill Panel Yield Forces on HBEs and VBEs Respectively. 63
xvi
4-4 Assumed HBE Axial Force Distribution Due to Horizontal Components
of Infill Panel Yield Forces on HBE 66
4-5 Structures with SPSW Implemented at Different Locations 67
4-6 Normalized Axial Force Distribution in Intermediate HBE
Due to Horizontal Components of Infill Panel Yield Forces on HBE 69
4-7 Decomposition of Loading on Intermediate HBE:
(A) Typical Intermediate HBE;
(B) Intermediate HBE Subjected to Infill Panel Yield Forces;
(C) Intermediate HBE Subjected to Plastic End Moments 71
4-8 Infill Panel Yield Forces on Simply Supported Intermediate HBE 72
4-9 Intermediate HBE End 73
4-10 Simplification of Infill Panel Forces on HBE Segment 74
4-11 Free Body Diagrams of Simply Supported HBE
under Fundamental Loading Due to Sub-Tension Fields 75
4-12 Free Body Diagram and Shear Diagram of Simply Supported Beam
under Partial Uniform Load 76
4-13 Free Body Diagram and Shear Diagram of Simply Supported Beam
under Partial Uniform Moment 77
4-14 Decomposition of Infill Panel Yield Forces
on the Simply Supported Beam and the Corresponding Shear Diagrams 79
4-15 Free Body Diagram, Moment Diagram and Shear Diagram of an HBE
with RBS Connections under Plastic End Moments Due to Frame Sway 83
xvii
Intermediate HBE Collapse Mechanisms Using Equilibrium Methods for:
(a) Vertical Components of Infill Panel Yield Forces;
(b) Left End Redundant Moment;
(c) Right End Redundant Moment;
(d) Combined Moment Diagram 87
4-17 Free Body Diagrams of Intermediate HBE
for Calculation of Moment Demand at VBE face 91
4-18 Yielding Patterns at RBS 99
4-19 Geometries of RBS Connections 100
4-20 Design Procedure of Intermediate HBEs Having RBS Connections 103
4-21 Design Procedure of Intermediate HBEs without RBS Connections 105
5-1 Typical Steel Plate Shear Wall and
Analogous Vertical Cantilever Plate Girder 110
5-2 Deformation of a Cantilever Plate Girder under Transverse Load
(Adapted from Wagner 1931) 111
5-3 Relationship between the Flexibility Factor
and the Stress Uniformity Ratio 114
5-4 Relationship between the Flexibility Factor
and the Stress Amplification Factor 116
5-5 Uniformity of Tension Fields (a) Pushover Curves,
(b) Schematic of Tension Fields, (c) Uniformity of Panel Stresses 118
5-6 In-plane Free Body Diagram of the VBE at the ith story
for Determination of Shear Demand 120
5-7 Deformation and Yield Patterns of SPSW2 after 6xδy
(from Lubell et al. 2000) 124
xviii
FIGURE TITLE PAGE
5-8 First-Story of Driver's SPSW (Photo: Courtesy of Driver. R.G.) 125
5-9 Yield Zone of VBE in SC Specimens (from Park et al. 2007) 126
5-10 Yield Zone of VBE in WC4T (from Park et al. 2007) 126
5-11 WC4T at the End of Test (from Park et al. 2007) 127
5-12 VBE Free Body Diagrams 129
5-13 Strong and Weak Axes of VBEs 132
5-14 Free Body Diagram of the VBE at the ith Story:
Case A Boundary Conditions 132
5-15 Deflection of VBE at Element Level 134
5-16 Interaction of Critical Loads for Out-of-Plane Buckling of VBE:
Case A Boundary Conditions 136
Case B Boundary Conditions 137
Case B Boundary Conditions 139
Case C Boundary Conditions 140
Case C Boundary Conditions 142
Case D Boundary Conditions 143
Case D Boundary Condition 145
5-23 Test Setup-UBC Test (from Lubell et al. 2000) 147
5-24 Load Deformation Curves for SPSW4:
(a) First Story; (b) Fourth Story
(from Lubell et al. 2000) 147
xix
5-25 Out-of-Plane Buckling of Bottom VBE
(Photo: Courtesy of Ventura. C.E.) 148
5-26 SPSW N at the End of the Tests and Hysteretic Curves 149
5-27 SPSW S at the End of the Tests and Hysteretic Curves 149
6-1 Plastic Mechanism of SPSWs (From Berman and Bruneau 2003) 154
6-2 Hystereses of a Single-Story SPSW
(Adapted from Berman and Bruneau 2005) 155
6-3 Test Results of a Multistory SPSW (Adapted from Driver et al. 1997) 156
6-4 Single-Story SPSW Example 159
6-5 The Relationship between Ωκ and κ (Assuming α=45º and η=1 ) 163
6-6 The Relationship between κbalanced and η (Assuming α=45º) 165
6-7 The Relationship between Aspect Ratio (L/h) and κbalanced (Assuming η=1) 165
6-8 Schematic of a Typical Multistory SPSW 166
6-9 Segments of a Uniformly Yielded Multistory SPSW 167
6-10 Description of Example Four-Story SPSW 169
6-11 Modified Design Forces of an Example Four-Story SPSW 170
6-12 Decompositions of Lateral Forces and SPSW System 172
6-13 Plastic Mechanism of Boundary Frame 174
6-14 SPSW Sub-Frames 175
6-16 Dual Strip Model and Ground Motion Information 179
6-17 Results from Time History Analyses 181
6-18 Typical Hysteretic Curves (from FEMA 450) 182
xxi
4-1 Parameters for Determining Shear Forces 77
4-2 Magnitudes of Top and Bottom Tension Fields
and Defining Parameters of Fish Plates 81
4-3 Magnitudes of HBE End Shears from Different Models 82
4-4 Summary of Cross-Section Properties and Flange Reduction Geometries 93
4-5 Design Forces at VBE face 94
4-6 Design Demands and Available Strengths at VBE Faces 107
5-1 Evaluation of VBE Shear Demand and Strength 122
5-2 Summary of VBE End Conditions 131
5-3 Evaluation of VBE Out-of-Plane Buckling 150
6-1 Summary of Design Story Shears and Infill Panel Thicknesses 177
6-2 Design Summary of Boundary Frame Members 177
xxiii
NOTATIONS
cid depth of column at story i
E young's modulus
ypif yield stress of infill panel at story i
H , sih story height
cI moment of inertia of column
oI moment of inertia of top flange of plate girder
uI moment of inertia of bottom flange of plate girder
yiI moment of inertia of column at story i taken from the weak axis
L bay width
botiM moment developed at the bottom of column at story i
PLM plastic moment at the left end of beam
PRM plastic moment at the right end of beam
topiM moment developed at the top of column at story i
sn number of SPSW stories
P axial compression in beam
DliP equivalent earthquake load applied at the left end of beam at story i
DmiP resultant force from uniform earthquake forces applied along beam at story i
DriP equivalent earthquake load applied at the right end of beam at story i
R response modification factor
xxiv
yR ratio of expected to nominal yield stress of boundary frame.
ypR ratio of expected to nominal yield stress of infill panel
ft thickness of beam flange
wt thickness of beam web
wcit thickness of column web at story i
wit thickness of infill panel of story i
designV lateral design force applied on SPSW
iV shear at story i
niV nominal shear strength of infill panel at story i
pV plastic strength of SPSW
cy compression portion of beam web
ty tension portion of beam web
Z plastic section modulus of beam
iα infill tension field inclination angle of story i
β cross-section plastic moment reduction factor
Lβ cross-section plastic moment reduction factor of the left column face
minβ minimum value of cross-section plastic moment reduction factor
Rβ cross-section plastic moment reduction factor of the right column face
Sβ plastic moment reduction factor at maximum beam interior span moment location
wβ ratio of the applied axial compression to the nominal axial strength of beam web
iδ arbitrarily selected nonzero column deflection factor
xΔ distance between RBS center and assumed plastic hinge
yΔ flange width difference between the RBS center and assumed plastic hinge
η RBS plastic section modulus reduction ratio
oη top flange deflection due to web tension actions in plate girder
uη bottom flange deflection due to web tension actions in plate girder
xxv
κ percentage of the total lateral design force assigned to infill panel
ν Poisson's ratio
cσ compression axial yield strength of beam web
maxσ maximum of the web tension force components paralleling with the stiffener of
plate girder
meanσ mean of the web tension force components paralleling with the stiffener of plate
girder
yσ vertical stress in beam web
y unσ − equivalent constant vertical stress in beam web
xyτ shear stress in beam web
tω flexibility factor
xbiω horizontal component of infill panel yield force along beam
xciω horizontal component of infill panel yield force along column
ybiω vertical component of infill panel yield force along beam
yciω vertical component of infill panel yield force along column
xxvii
ABBREVIATIONS
ATC Applied Technology Council
CSA Canadian Standards Association
HBE Horizontal Boundary Element
(Buffalo, NY)
NCREE National Center for Research on Earthquake Engineering (Taipei, Taiwan)
NEHRP National Earthquake Hazards Reduction Program
PGA Peak Ground Acceleration
RBS Reduced Beam Section
UB University at Buffalo
VBE Vertical Boundary Element
1.1 General
Steel Plate Shear Walls (SPSWs) consist of infill steel panels surrounded by columns,
called Vertical Boundary Elements (VBEs), and beams, called Horizontal Boundary
Elements (HBEs).These infill panels are allowed to buckle in shear and subsequently
form diagonal tension fields when resisting lateral loads. Energy dissipation of SPSWs
during seismic events is principally achieved through yielding of the panels along the
diagonal tension fields.
Consistent with capacity design principles, the Canadian Standard S16 on Limit State
Design of Steel Structures (CSA 2000) and the AISC Seismic Provisions for Structural
Steel Buildings (AISC 2005) require the boundary frame members (i.e. HBEs and VBEs)
to be designed to be sufficiently rigid to ensure the development of infill tension fields
and remain elastic when the infill panels are fully yielded, with exception of plastic
hinges at the ends of HBEs and at the VBE bases that are needed to develop the expected
plastic mechanism of the wall when rigid HBE-to-VBE and VBE-to-ground connections
are used.
However, recent testing on multi-story SPSWs having reduced beam section (RBS)
connections by Qu and Bruneau (2008) revealed that the yielding pattern of RBS
connections in SPSWs is quite different from that of beams in conventional steel moment
frames. Moreover, the intermediate HBEs of their specimen ultimately failed due to
fractures at the VBE faces; however, no factures developed in the reduced beam flange
regions. Note that intermediate HBEs are those to which are welded steel plates above
and below, by opposition to anchor HBEs that have steel plates only below or above. It
would be important to investigate the reasons for the difference in observed yielding
behavior and to develop an improved capacity design procedure for HBEs to better
ensure the ductile performance of SPSWs.
2
In addition, recent experimental data allow to revisit the effectiveness of equations in
current design codes that are derived from plate girder theory and adopted with the
intention of ensuring an adequate flexibility for VBE design. Furthermore, additional
work is necessary to investigate the relative contribution of boundary frame moment
resisting action to the overall strength of SPSWs and possibly achieve an optimum design
of SPSWs accounting for that contribution.
1.2 Scope and Objectives
This report presents analytical work conducted to investigate the behavior and design
procedure of horizontal and vertical boundary frame members that may impact SPSW
performance.
First, new analytical models are developed to calculate the HBE plastic moment
accounting for the reduction in strength due to the presence of biaxial internal stress
conditions, and to revisit and develop improved capacity design procedure for HBEs
taking into account these reduced plastic moments of HBEs. These advances make it
possible to investigate and explain the intermediate HBE failure observed in recent tests
(Qu and Bruneau, 2008).
Next, the adequacy of a flexibility limit for VBE design specified by the current design
codes is assessed using new analytical models developed to prevent the undesirable in-
plane and out-of-plane performances of VBEs.
Furthermore, a SPSW design procedure accounting for the contribution of boundary
frame moment resisting action is proposed.
1.3 Outline of Report
Section 2 contains a brief overview of past analytical research related to this structural
system in applications to provide earthquake resistance.
Section 3 begins with a discussion of HBE plastic moment accounting for the presence of
axial force, shear force, and vertical stresses in HBE web due to infill panel yield forces.
3
Based on the results derived from Section 3, Section 4 develops a revised capacity design
procedure for HBEs using enhanced free body diagrams and principle of superposition.
The new developed model can be used to explain previously observed undesirable HBE
performance.
Section 5 assesses the adequacy of flexibility limit for VBE design specified by the
current design codes. Derivation of a flexibility factor in plate girder theory and how that
factor was incorporated into current codes are reviewed, followed by the development of
analytical models for preventing shear yielding and estimating out-of-plane buckling
strength of VBEs of SPSWs.
Building on the knowledge developed in the prior sections for boundary frame member
behavior and design, Section 6 investigates the contribution of boundary frame moment
resisting action to the overall wall strength. A balanced design procedure is developed to
account for this action followed by the derivation of three different procedures for the
SPSWs having weak infill panels.
Finally, summary, conclusions, and recommendations for future research in SPSWs are
presented in Section 7.
2.1 Introduction
Prior to key research performed in the 1980s (Thorburn et al. 1983), designs of SPSW
infill panels only allowed for elastic behavior, or shear yielding in the post-elastic range.
This design concept typically resulted in selection of relatively thick or heavily-stiffened
infill panels. While resulting in a stiffer structure that would reduce displacement demand
as compared to the bare steel frame structure during seismic events, these designs would
induce relatively large infill panel yield forces on the boundary frame members, resulting
in substantial amounts of steel used and expensive detailing. Numerous experimental and
analytical investigations conducted since 1980s have demonstrated that a SPSW having
unstiffened thin infill panels allowed to buckle in shear and subsequently form a diagonal
tension field absorbing input energy, can be an effective and economical option for new
buildings as well as for the retrofit of existing constructions in earthquake-prone regions.
While extensive reviews of past research can be found in the literature (i.e. Berman and
Bruneau 2003, Vian and Bruneau 2005, Sabelli and Bruneau 2007, to name a few), some
work relevant to the work presented here is summarized below, with more emphasis on
analytical studies using various modeling strategies.
2.2 Thorburn, Kulak, and Montgomery (1983)
Based on the theory of diagonal tension field actions first proposed by Wagner (1931),
Thorburn et al. (1983) investigated the postbuckling strength of SPSWs and developed
two analytical models to represent unstiffened thin infill panels that resist lateral loads by
the formation of tension field actions. In both cases, contribution to total lateral strength
from the compressive stresses in the infill panels were neglected because it was assumed
that plate buckles at a low load and displacement level. In addition, it was assumed that
the columns were continuous over the whole height of the wall, to which the beams were
connected using simple connections (i.e. "pin" connections).
6
The first model, an equivalent brace model used to provide the story stiffness of a panel,
represents the infill panels as a single diagonal tension brace at each story. Based on
elastic strain energy formulation, analytical expressions were provided for the area of this
equivalent brace member for two limiting cases of column stiffness, namely, infinitely
rigid against bending and completely flexible.
The second model proposed by Thorburn et al. (1983) is strip model (also known as a
multi-strip model), in which each infill panel is represented by a series of inclined pin-
ended only members, as shown in figure 2-1, that have a cross-sectional area equal to
strip spacing times the panel thickness.
FIGURE 2-1 Schematic of Strip Model (Thorburn et al. 1983)
It was found that a minimum of ten strips is required at each story to adequately replicate
the behavior of the wall. Using the principle of least work, the inclination angle for the
strip, denoted as α , and equal to that of the tension field can be determined for the
infinitely rigid column case, as,
7
(2-1)
where H is the story height; L is the bay width; t is the infill panel thickness; and bA
and cA are cross section areas of the beam and column, respectively.
2.3 Timler and Kulak (1983)
Based on the work by Thorbrrn et al. (1983), Timler and Kulak (1983) considered the
effects of column flexibility and revised equation (2-1):
4 3
1 2tan
11 360
α
(2-2)
where cI is the moment of inertia of column, and all other terms were defined previously.
This equation appears in both the Canadian CSA-S16-01 Standard (CSA 2000) and the
2005 AISC Seismic Provisions (AISC 2005) for design of SPSWs.
2.4 Driver, Kulak, Kennedy and Elwi (1997)
Driver et al. modeled their large-scale four-story SPSW specimen tested under cyclic
loading, considering both FE model and strip model.
The FE model, as shown in figure 2-2 used quadratic beam elements to represent the
beams and columns, and quadratic plate/shell elements to model the infill plates.
8
FIGURE 2-2 FE Model of Test Specimen (Driver et al. 1997)
As-built dimensions and measured material properties were included into the model. The
FE simulation reasonably predicted the ultimate strength for all stories. However, the
model overestimated the stiffness of the specimen. It was concluded that this discrepancy
was due to the inability to include the geometric nonlinearities.
Using the procedure originally presented by Thorburn et al. (1983), Driver et al. (1997)
also developed the strip model of their specimen, as shown in figure 2-3, and performed a
pushover analysis to calculate the envelope of cyclic curves experimentally obtained.
Inelastic behavior in the inclined tension field strips and the frame members was
modeled. Although the model slightly underestimated the elastic stiffness of the test
specimen, excellent agreement was obtained with the experimentally observed ultimate
strength.
9
FIGURE 2-3 Strip Model of Driver et al. Specimen (Driver et al. 1997)
2.5 Berman and Bruneau (2003)
Based on the strip model, Berman and Bruneau (2003) performed plastic analysis to
explore the behavior of SPSWs. Fundamental plastic collapse mechanism were described
for single story and multistory SPSWs with either simple or rigid beam-to-column
connections.
For a single story SPSW with simple beam-to-column connections, as shown in figure 2-
4, Berman and Bruneau (2003) demonstrated, using equilibrium and kinematic methods
of plastic analysis, that the assumed collapse mechanism for the strip model produces an
expression for story shear strength, identical to that of the CAN/CSA S16-01 procedure
used in calculating the shear resistance of an SPSW infill panel.
Berman and Bruneau further extended plastic analysis of SPSWs to the multistory cases.
They examined two types of plastic mechanisms, namely, a uniform collapse mechanism
10
and a soft-story collapse mechanism which are shown in figure 2-5. These mechanisms
and their corresponding ultimate strengths, provide the engineer simple equations for
estimating the ultimate capacity of a multistory SPSW and preventing the possible soft
story mechanism. The ultimate strengths predicted for these plastic mechanisms were
validated by the experimental data from past research.
FIGURE 2-4 Single Story SPSW Collapse Mechanism
(Berman and Bruneau 2003)
FIGURE 2-5 Example of Plastic Collapse Mechanism for Multi-story SPSWs
(Berman and Bruneau 2003)
2.6 Behbahanifard, Grondin and Elwi (2003)
Behbahanifard et al. (2003) generated a FE model to model their three-story SPSW
specimen. Their model, as shown in figure 2-6, was developed based on the nonlinear
dynamic explicit formulation, implementing a kinematic hardening material model to
simulate the Bauschinger effect, after experiencing convergence problems analyzing the
model using the implicit FE model formulation.
FIGURE 2-6 FE Model of Test Specimen (Behbahanifard et al. 2003)
Excellent agreement was observed between the test results and the numerical predictions.
Using the validated FE model, parametric studies were conducted to identify and access
some of the non-dimensional parameters affecting the behavior of a single panel SPSW
with rigid floor beams and subjected to shear force and constant gravity loading.
The researchers found that lower aspect ratio results in greater strength and non-
dimensional lateral stiffness for SPSWs. However, this effect is negligible within the
aspect ratio range of 1.0 to 2.0, but noticeable for aspect ratio less than 1.0.
12
In addition, as the column lateral stiffness increased relative to the panel stiffness, the
shear wall capacity approached the yield capacity. The stiffer column can more
effectively anchor the tension field resulting from infill panel yielding, therefore allowing
a more efficient use of the material composing the system.
It was demonstrated that initial out-of-plane imperfections in the infill panel could have a
significant influence on the stiffness of the shear panel, while they have no effects on the
ultimate shear strength.
Furthermore, the effects of overturning moment and applied gravity load were also
investigated using the validated FE model. It was found that increasing either gravity load
or the overturning moment reduces the elastic stiffness of the shear wall panel in an
almost linear manner and also significantly reduces the normalized capacity and ductility.
2.7 Kharrazi, Ventura, Prion, and Sabouri-Ghomi (2004)
Kharrazi et al. (2004) proposed a numerical model referred to as the Modified Plate-
Frame Interaction (M-PFI) model to analyze the shear and bending of ductile SPSW. The
objective was to describe the interaction between those components and characterize the
respective contribution to deformation and strength at whole structure level. Thus, the M-
PFI model separates the behavior of ductile SPSW into three parts: elastic buckling, post-
buckling, and yielding. Several steps were involved in developing these equations. First,
a shear analysis was conducted that looked at the behavior of the infill panels and frames
respectively and then the shear-displacement relationships for each were superimposed to
obtain the shear behavior of the ductile SPSW. Second, a bending analysis was conducted
assuming that the frame and plate act as wide flange shape. Equations were proposed to
obtain certain points on a shear-displacement relationship that can be used for analyzing
the behavior of the wall.
Kharrazi et al. (2004) used test data from Driver et al. (1997) to evaluate the M-PFI
model using an assumed tension field inclination of 45 degree. The model overestimated
the initial stiffness by 5% and underestimated the ultimate capacity of the specimen by
about 10%, although it overestimated the specimen capacity slightly at initial yielding. It
13
should be mentioned that the model does not describe the ductility of the SPSW specimen
or the actual mechanism, nor does it provide a means of determining the frame forces for
use in design.
2.8 Vian and Bruneau (2005)
Vian and Bruneau (2005) investigated some new methods for the design of SPSW anchor
beams. Based on simple free body diagrams as shown in figure 2-7, and the principle of
superposition, they proposed a procedure to ensure that frame plastic hinging occurs in
the beams and not in columns. Limits were proposed to estimate the drift of a frame with
and without SPSW panels at yielding, so that the system may be assessed in the context
of the structural "fuse" concept, and considered to dissipate input energy through SPSW
panel yielding at a drift level less than that causing the frame members to yield.
MR
ML
MR
x
ω
ML
L
8
14
Lopez-Garcia and Bruneau (2006) performed some preliminary investigation on the
seismic behavior of intermediate beams in SPSWs using strip model. Of primary interest
was the determination of the strength level needed to avoid the formation of in-span
plastic hinges. To attain this objective, the seismic response of several SPSW models
designed according to the FEMA/AISC regulations and the Canadian standard CAN/CSA
S16-01 were analyzed by performing linear and nonlinear analysis. The SPSW models
considered in this research were designed as an alternative to provide the lateral load
resisting system to the four-story MCEER Demonstration Hospital, a reference building
model used as part of a broader MCEER research project. The intermediate beams of the
FEMA/AISC models were designed according to three different criteria: (I) for gravity
loads only; (II) for the ASCE 7 load combinations: (III) for the forces generated by fully
yielded webs.
In all cases, the thickness of the thinnest hot-rolled plate available turned out to be larger
than the minimum thickness required by the FEMA/AISC guidelines, which resulted in
SPSW models having the same plate thickness at all stores (constant plate thickness
case). In order to obtain insight into the behavior of intermediate beams of SPSWs having
different plate thickness at different stories, models whose plates have the minimum
thickness required were also considered for each of the abovementioned design criteria
(variable plate thickness case).
Seismic loads were calculated per FEMA 450 assuming that the structure was located in
Northridge, California ( 1.75SS g= and 0.75SS g= ). The SPSWs were analyzed using
the commercial computer program SAP-2000 version 8.3.3. Each infill panel was
modeled by a set of 10 parallel, uniformly spaced tension-only strips pinned at both ends
(i.e. elements capable of resisting tension axial force only), while the beams and columns
were modeled by conventional frame elements.
Forces imposed by the ASCE 7 load combination were assessed through FEMA 450's
Equivalent Lateral Force Procedure (linear static analysis). The inelastic behavior of the
SPSW models under seismic loading was analyzed by performing FEMA 450's Nonlinear
15
Static procedure (Pushover analysis). For this, P-M plastic hinges were modeled at the
ends of the beams and columns. The strain hardening ratio was set equal to 0.5%. Axial
plastic hinges having an elastic-perfectly plastic force-deformation relationship were
modeled at the middle of each strip. As indicated by FEMA 450, a set of gravity loads
equal to 0.25D L+ was applied to the models prior to the incremental application of
earthquake loads. The pattern of seismic forces was set equal to that indicated by the
Equivalent Lateral Force Procedure.
It was found that the FEMA/AISC models designed according to criteria I and II
developed in-span plastic hinges in intermediate beams, which is not allowed by the
FEMA/AISC regulations. Models designed according to criteria III exhibited the desired
behavior (i.e. inelastic deformations occur only at the ends of the beams and in the webs)
even when a global collapse mechanism developed. The behavior of CAN/CSA S16-01
models was found to be satisfactory at the response level corresponding to the design
basic earthquake, but the global mechanism of these models included in-span plastic
hinges in the anchor beams. All these observations were found to apply regardless of
whether the SPSW systems have constant or variable plate thickness.
2.10 Shishkin, Driver and Grondin (2005)
Shishkin et al. (2005) proposed refinements to the strip model, as described by Thorburn
et al. (1983), to obtain a more accurate prediction of the inelastic behavior of SPSW
using a conventional structural engineering software package. The refinements were
based on observations from laboratory tests on SPSW specimens. Modeling efficiency
was also evaluated against accuracy of the solution. A modified version of the strip
model was proposed as pictured in figure 2-8, which was shown to be efficient while
maintaining a high degree of accuracy. In this model, a pin-ended compression strut was
used to account for the small contribution of the stiffness and the strength of the infill
panel from compressive resistance, which may be significant in the corner regions where
effective length of the plate under compression is small.
16
FIGURE 2-8 Schematic of the Modified Strip Model (Shishkin et al. 2005)
The parameters of the proposed model are generic and can be implemented into structural
analysis program with pushover analysis capacities. A parametric study was also
performed to determine the sensitivity of the predicted nonlinear behavior to variations in
the angle of inclination of the infill panel tension field.
2.11 Purba and Bruneau (2006)
As part of the further investigations on some concerns reported by Vian and Bruneau
(2005), Purba and Bruneau (2006) performed some analytical study on the behavior of
unstiffened thin SPSW having openings on the infill plate under monotonic pushover
displacement. Two SPSW systems with openings proposed by Vian and Bruneau (2005),
namely the perforated and the cutout corner SPSW, were discussed.
The researchers first studied individual perforated strips, which may play an important
role in the behavior of perforated SPSW. The effect of mesh refinement on the
17
idealizations. The results were presented in terms of stress-strain distribution throughout
the strip section as well as in terms of global deformations. Then, a series of one-story
SPSWs having multiple perforations on panels were considered, with variation in
perforation diameter, boundary conditions, infill plate thickness, material properties
idealization, and element definition. The objective of this analysis was to verify the
accuracy of the results obtained from FE analysis of individual perforated strips to predict
the SPSW strength by summing the strength of "simpler" individual strips. Shell elements
were used to model the infill plates as well as the boundary frame member webs and
flanges. Good agreement in overall behavior between the models considered and
individual perforated strip model was observed. The applicability of the equation
proposed by previous researchers to approximate the strength of a perforated panel was
also re-assessed.
In addition, two cutout corner SPSW models were also investigated. The first model
replicated the cutout corner SPSW specimen tested by Vian and Bruneau (2005) in which
a flat-plate reinforcement was introduced along the cutout edges. The second model
considered had a T-section reinforcement along the cutout edges; this model was built by
adding a new plate perpendicularly to the previous flat-plate reinforcement. No
significant difference between the two models was observed in terms of frame
deformations and stress distributions along the cutout corner SPSW. However, some
local effects were observed adjacent to the cutout corner, in terms of diagonal
displacement of the cutout reinforcement plate and stress distribution along the length of
the plates.
Berman and Bruneau (2008) performed research on developing capacity procedure for
VBEs in SPSWs. They reviewed the current approaches provided in the AISC Seismic
Provisions (AISC 2005) for determination of capacity design loads for VBEs of SPSWs
and identified the deficiency of these procedures. Then, a new procedure was proposed
18
that based on a fundamental plastic collapse mechanism and linear beam analysis to
approximate the design actions for VBEs for the given infill panels and HBEs. The VBE
free body diagrams they used were shown in figure 2-9. Their procedure does not involve
nonlinear analysis, making it practical for use in design. VBE design loads were
estimated using the proposed procedure for two example SPSW configurations. It is
found that VBE design forces predicted from the proposed procedure agree well with
those from the nonlinear pushover analysis.
FIGURE 2-9 VBE Free Body Diagrams (Berman and Bruneau 2008)
19
3.1 Introduction
The HBE yield patterns observed in the recent tests on SPSWs by Qu et al. (2008)
indicate that the stress distribution at the HBE plastic hinges is more complex than
accounted by simple analysis methods. Therefore, if one wishes to understand how to
design an HBE, the first step is to correctly quantify the strength of HBE plastic hinges
accounting for this condition.
Conventional plastic analysis procedures for determining the plastic moment of wide
flange member in a moment frame can not be applied to HBEs of SPSWs because the
HBE web is under large bi-axial stress condition. To account for this effect, analytical
procedures to estimate the plastic moment resistance of HBEs subjected to axial force,
shear force, and vertical stresses due to infill panel forces, are needed to ensure
predictable and ductile behavior of HBEs.
This section first discusses the loading characteristics of HBE cross-sections. Following a
review of the conventional procedure to calculate the plastic moment of wide flange
members in a conventional steel moment frame, the reduced yield strength of HBE web
under shear and vertical stresses is studied using the von Mises yield criterion. Analytical
procedures for estimating the plastic moment of intermediate HBEs and anchor HBEs are
then developed, respectively. Note that intermediate HBEs are those to which are welded
steel plates above and below, by opposition to anchor HBEs that have steel plates only
below or above. Procedures for calculating plastic moment of intermediate HBEs under
equal and unequal top and bottom tension fields are first investigated. Those procedures
are verified by FE examples and are simplified for practical purpose. Next, additional
numerical examples are developed to confirm that those procedures proposed for
intermediate HBEs can also apply to anchor HBEs. Note that the analytical procedures to
estimate HBE plastic moment developed in this section will be used in Section 4 for
capacity design of HBEs.
3.2 Loading Characteristics of HBE Cross-Sections
In any multistory SPSW such as the one shown in figure 3-1, the HBEs can be
differentiated as anchor HBEs and intermediate HBEs. Anchor HBEs are the top and
bottom ends of a SPSW and that anchor the SPSW infill panel yield forces. Since they
are loaded by infill tension field forces only on one side (either above or below), they are
typically of substantial size. Intermediate HBEs are the beams at all other levels. The
variation between the top and bottom infill panel stresses acting on the HBE can
sometimes be small or null when the top and bottom infill panels of identical (or near
identical) thicknesses are both yielding. Comparing with the sizes of anchor HBEs,
intermediate HBEs are often relatively small.
A
A
B
B
D
D
C
C
F
F
E
E
FIGURE 3-1 Typical Multistory SPSW
To understand the infill panel effects on HBE behaviors, consider the aforementioned
SPSW with rigid HBE-to-VBE connections. The shear stress and external loading at the
HBE ends are schematically shown in figure 3-2, where τ represents the shear stress in
HBE web; and P and M represent the axial force and moment acting at HBE ends,
21
respectively; when the expected plastic collapse mechanism of the SPSW develops. In
this figure, 1ybiω + and ybiω represent the vertical components of the top and bottom infill
tension fields, respectively. Mathematical expressions for 1ybiω + and ybiω will be given in
the Section 4.
22
Note that an HBE is typically in compression which will be demonstrated in Section 4.
The magnitudes of τ, P, and M vary at different locations along an HBE. Although the
direction of shear stress, τ, depends on the resulting shear effects due to tension field
forces and HBE flexural actions, it has no effect on HBE plastic moment resistance as
demonstrated in Section 3.5.3. Accordingly, the loading characteristics of HBE cross-
sections are summarized in table 3-1. Note that for the purpose of the present discussion,
flexure designated as "+"or "-" respectively refers to the bending action producing
tension or compression in the flange on which the greater tension field force is applied.
For intermediate HBEs with equal top and bottom tension fields, the acting direction of
flexure has no impact on the plastic moment resistance as demonstrated in Section 3.5.1.
TABLE 3-1 Summary of Loading Characteristics of HBE Cross-Section
HBE type Corresponding
Intermediate C/D 1 Intermediate C √ 1> Intermediate D √ 1>
Anchor A √ 0-a Anchor B √ 0-a Anchor E √ ∞-b Anchor F √ ∞-b
-a when 0yiω = . -b when 1 0yiω + = .
3.3 Plastic Moment of Wide Flange Members in Moment Frame
A well-known lower bound approach to estimate plastic moment for wide flange
members in steel moment frame, based on stress diagrams, and classic plastic analysis
can be used to account for the combined interaction of flexure, axial and shear forces
(Bruneau et al. 1998). Using this procedure, the uniform shear stress, wτ , assumed to act
on the web of the cross-section as a result of the applied shear force, V , is calculated as:
w w w
V h t
23
where wh and wt are the web depth and thickness of the cross-section, respectively.
Then, the von Mises yield criterion is used to calculate the maximum axial stress, wσ ,
that can be applied on the web (i.e., the remaining axial yield strength):
2 23w y wfσ τ= − (3-2)
where yf is the yield strength of steel.
Strength of the flanges remains yf . In the case that the neutral axis remains in the web; as
shown in figure 3-3, location of the neutral axis, oy , is given by:
o w w
where P is the applied axial compression.
Neglecting strain hardening, the plastic axial stress diagram of a typical wide flange
section can be divided into pure flexural and axial contributions as shown in figure 3-3.
The contributions of the web and flanges to the plastic moment resistance can be
( ) ( )
t d t t y M σ σ−
− = − (3-4)
( ),P V pr flange f f f yM b t d t f− = − (3-5)
, , ,P V P V P V pr pr web pr flangeM M M− −= + (3-6)
where the superscript (P, V) indicates that the plastic moment is reduced taking into
account the applied axial and shear forces, and where fb and ft are the flange width and
thickness of the cross-section, respectively.
24
Wide flange section Shear stress Axial stress Flexure Axial compression
= + yo
bf
tw
tf
+σw
+fy
-σw
+σw
+fy
-σw
-fy
FIGURE 3-3 Example of Plastic Resistance of a Wide Flange Structural Shape
Subjected to Flexure, Axial, and Shear Forces
The above lower bound approach, which provides acceptable results, has been used for
steel moment frame design. However, for SPSW, the experimental results described in
Qu et al. (2008) provide evidence that a more sophisticated procedure is warranted. To do
so, a review of the von Mises criterion in plane stress condition is necessary and done in
the following section.
3.4 Reduced Yield Strength in HBE Web
To better understand the reduced axial yield strength in HBE web accounting for the
shear stress and vertical stresses due to infill panel forces, two elements are arbitrarily
selected from the axial tension and compression zones in the web of a typical
intermediate HBE segment as shown in figure 3-4. These elements are in plane stress
condition. The free body diagrams, stress diagrams and Mohr’s circles are also
schematically shown in figure 3-4. Note that the exact Mohr’s circles of elements A and
B depend on the magnitudes of the vertical components of the tension fields, i.e.
1ybiω + and ybiω , applied above and below the intermediate HBE segment.
25
O O
FIGURE 3-4 Loading of Intermediate HBE Segment and
Mohr's Circles of the Elements on the Web
The results of classic mechanics of materials show that the radius of the Mohr’s circle
can be determined as:
(3-7)
Graphically, the maximum and minimum in-plane principal stresses, σ1 and σ2, can be
calculated as:
σ σ σ
σ σ σ
In addition, for steel subjected to multi-axial plane-stress conditions and having a
uniaxial elasto-perfectly plastic behavior, steel can be modeled using the von Mises yield
criterion and expressed as:
1 2 yfσ σ σ σ − + + = (3-10)
Substituting(3-7), (3-8) and (3-9) into (3-10) and normalizing the resulting equation, the
von Mises yield criterion can be rewritten as:
2 2 2
3 1y y xyx x
− + + =
(3-11)
Equation (3-11) can be reorganized as a quadratic equation in terms of x yfσ or xy yfτ
and solved to respectively give the reduced axial and shear yield strengths for given
values of the other terms:
2 2
y y xyx
= ± − −
xy y yx x
= ± − + −
(3-13)
Note that the reduced tension and compression axial yield strength can be obtained by
considering the "+" and "-" cases in (3-12) respectively.
Equations (3-12) and (3-13), can be simplified as shown in (3-14) and (3-15) for cases of
uniaxial stress and pure shear respectively.
27
2
= ± − =
f τ σ σ= = = (3-15)
Note that (3-14) is equivalent to (3-2) which is used for estimating the plastic moment
resistance of wide flange members in steel moment frame. The von Mises yield criterion,
in the x yfσ and xy yfτ space, for given values of y yfσ are shown in figure 3-5 as a
graphical representation of (3-12) .
τxy
fy
-1.20
-1.00
-0.80
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
1.20
/fy σy =0.6
/fy σy =0.4
/fy σy =0.8
FIGURE 3-5 Reduced Yield Strength per the Von Mises Criterion in Plane Stress
28
As observed in (3-12), the reduced axial yield strength is a function of shear stress xyτ
and vertical stress yσ . In the absence of vertical axial stresses (i.e. 0yσ = ), the
compression and tension axial yield strengths are of the same magnitude. This is the case
illustrated in figure 3-3 , for a wide flange member in conventional moment frame.
However, as shown in figure 3-5, the presence of vertical stresses results in unequal
tension and compression axial yield strengths. This is the stress condition found in an
HBE cross-section. Note from figure 3-5 that the compression axial yield strength is
significantly reduced, and can even become null under the combined action of shear and
vertical stresses. The tension axial yield strength is also affected by the shear and vertical
stresses, but to a lesser degree. Reading the figure differently, due to the presence of
given vertical and axial compression/tension stresses, the shear yield strength is also
reduced. Given that the vertical stresses in the whole HBE web may not be constant,
figure 3-6 is helpful to compare the compression and tension axial yield strengths for a
wide range of vertical stress ratio (i.e. y yfσ ). Note from figure 3-6 that the maximum
value of y yfσ can be reached is less than 1.0 in some cases (e.g. for the case of
0.40xy yfτ = ). It is because the maximum allowable vertical stress is reduced due to the
presences of the axial stress in horizontal direction (i.e. xσ ) and the shear stress (i.e. xyτ )
as shown in figure 3-5.
29
3.5 Intermediate HBE under Equal Top and Bottom Tension Fields
The plastic moment resistance of intermediate HBE under constant vertical stresses,
resulting from equal top and bottom tension fields, is first discussed because this case
provides some of the building blocks necessary to understand the more complex
scenarios presented later in this section.
3.5.1 Derivation of Plastic Moment
In the spirit of classic plastic cross-section analysis and following a procedure similar to
that presented in Section 3.3, also using reduced axial yield strength obtained from the
von Mises yield criterion accounting for the vertical and shear stresses as presented in
Section 3.4, one can generate the stress diagrams for a fully plastified wide flange section
under uniform vertical tension fields as shown in figure 3-7. Note that this case
corresponds to the stress conditions generated in an intermediate HBE when the infill
panels above and below the HBE are of equal thicknesses and fully yielded (i.e.
1ybi ybiω ω += ). Also note that all the equations derived in this section remain valid for case
of opposite flexure case (i.e. the top and bottom parts of figure 3-7 show the stress
diagrams corresponding to both flexure cases.).
As traditionally done in structural steel for wide flange sections, uniform shear stress is
assumed to act on the HBE web. In addition, a constant vertical tension stress is assumed
in the HBE web as a result of the identical top and bottom tension fields. The shear stress
is calculated according to (3-1). The uniform vertical stress is given by:
ybi y
wt ω
σ = (3-16)
Consistent with (3-12), the tension and compression axial yield strengths, tσ and cσ , can
be respectively calculated as:
y yt w
= + − −
y yc w
= − − −
-σc
τw
hw
bf
tw
tf
+σt
ωybi
yc
yt
-fy
-σc
τw
hw
bf
tw
ωybi
yc
yt
FIGURE 3-7 Stress Diagrams of Intermediate HBE Cross-Section
under Flexure, Axial Compression, Shear Force, and Vertical Stresses due to Equal Top and Bottom Tension Fields
From the geometry in figure 3-7,
t w cy h y= − (3-19)
where cy and ty represent compression and tension portion of the web ,respectively.
For the most common case that the neutral axis remains in the web, the governing
equation of axial force equilibrium can be given as
c c w t t wy t y t Pσ σ+ = (3-20)
32
where P is the applied axial compression.
The contribution of the flanges to axial compression is null since the resultant axial forces
of each flange have the same magnitude but opposite signs (and thus cancel each other).
Substituting (3-19) into (3-20) and solving for cy gives,
t w
(3-21)
where wβ is the ratio of the applied axial compression to the nominal axial strength of the
web, which is given by:
w y w w
P f h t
β −= (3-22)
The contributions of the web and flanges to the plastic moment resistance, pr webM − and
pr flangeM − , can be respectively determined as:
2 2 2 2 w t w c
= − − −
(3-23)
( )pr flange y f f fM f b t d t− = − (3-24)
A cross-section plastic moment reduction factor, β , is defined to quantify the loss in
plastic strength attributable to the combined effects of the applied axial compression,
shear force, and vertical stresses due to the infill panels forces acting on the HBE. This
factor, β , can be determined as:
pr web pr flange
where Z is the HBE plastic section modulus.
For the extreme case for which the web of the cross-section makes no contribution to the
flexure, the minimum value of the reduction factor, minβ , is obtained, and is given as:
min pr flange
β −= (3-26)
This will happen when the entire web is under uniform compression.
3.5.2 FE Verification
To validate the approach developed in Section 3.5.1 to calculate the cross-section plastic
moment reduction factor of the intermediate HBE under equal top and bottom tension
fields, a series of FE analyses were performed on a segment of HBE. The FE model is
shown in figure 3-8.
FIGURE 3-8 FE Model of Intermediate HBE Segment
under Flexure, Axial Compression, Shear Force, and Vertical Stresses due to Equal Top and Bottom Tension Fields
In this case, a W21x73 member was modeled in ABAQUS/Standard. The length of the
member was twice the cross-section depth. Material was assumed to be A572 Grade 50
steel with elasto-perfectly plastic constitutive behavior. Shell element (ABAQUS element
S4R) was employed for the web and flanges. A fine mesh with 9,000 elements (2,000
elements per flange and 5,000 elements for the web) was used in this model.
34
As shown in figure 3-8, uniformly distributed loads, 1ybiω + and ybiω , of identical
magnitude but applied in opposite directions at the top and bottom edges of the HBE web
represented the vertical components of the top and bottom tension fields. Uniform shear
stresses, xyτ , were applied along the HBE web edges. Axial forces applied at the ends of
the member represented the axial compression in the HBE.
The FE analysis was conducted in two stages to correctly replicate boundary conditions
and to achieve the desired load scenario while keeping the whole model in self-
equilibrium. In the first stage, the aforementioned tension field forces, shear stresses and
axial forces were applied on the FE model. In the next stage, a displacement controlled
analysis procedure was used to obtain the plastic moment resistance of the cross-section,
in which the rotations at the end of the HBE segment were proportionally increased up to
a magnitude of 0.035 rad, but in opposite directions.
A series of analyses were conducted on the FE model to assess the accuracy of the
analytical procedure. Results are shown in figure 3-9. In these analyses, for the given
axial compression and shear, the magnitudes of the identical tension fields were increased
from null to the maximum allowable value in the HBE web. To capture the entire range
of the solutions developed analytically, for each case of wβ , five different vertical stress
conditions (approximately equally spaced in the allowable range of y yfσ ) were
considered in the FE models. Note that the combined axial compression, shear and
vertical stresses selected for these analyses may not necessarily develop in a real
intermediate HBE. Also, note that the plastic moment reduction factor shown in figure
3-9 for the intermediate HBE cross-section reduces to that which corresponds to values
for the ordinary beam action in the absence of tension fields from the SPSW infill panels.
As shown in figure 3-9, the cross-section plastic moment reduction factors obtained from
the FE analysis agree very well with those predicted using the approach presented in
Section 3.5.1. One can also observe the cross section plastic moment reduction factor
varies from unity to a minimum when increasing the axial force, shear, and vertical
stress.
35
0.00 0.20 0.40 0.60 0.80 1.00 σyi/fy (σyi/fy=σyi+1/fy)
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
β
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
β
FIGURE 3-9 Plastic Moment Reduction Factor of Intermediate HBE Cross-Section under Axial Compression, Shear Force, and Uniform Vertical
Stresses: Analytical Predictions versus FE Results
36
3.5.3 Effects of Shear Direction
Though (3-17) and (3-18) show that the acting direction of shear stresses has no effects
on the resulting compression/tension axial yield strength, and consequently make no
difference in the calculated cross-section plastic moment reduction factor, numerical
examples were conducted to confirm this point. Directions of the edge shear shown in
figure 3-8 were switched and the FE model was reanalyzed for the cases shown in figure
3-9 case (b). The results from this new model (i.e.FE2) are plotted in figure 3-10 together
with those from the original model (i.e.FE1) and from the approach proposed in Section
3.5.1. As shown, the direction of shear stresses has no impact on the cross-section plastic
moment reduction factor.
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
β
Analytical βw=0.00
Analytical βw=0.40
Analytical βw=0.60
FE1 βw=0.00
FE1 βw=0.40
FE1 βw=0.60
FE2 βw=0.00
FE2 βw=0.40
FE2 βw=0.60
FIGURE 3-10 Effects of Shear Direction on Cross-Section Plastic Moment Reduction Factor of Intermediate HBE
3.6 Intermediate HBE under Unequal Top and Bottom Tension Fields
In Section 3.5, it was assumed that identical tension fields were acting above and below
the intermediate HBE. However, in many instances this is not the case and it is necessary
37
to investigate how the HBE plastic moment resistance would be affected by the unequal
top and bottom tension fields. As listed in table 3-1, both positive and negative flexure
cases are considered. The same fundamental concepts and modeling assumptions
previously presented still apply, except that a linear variation of vertical stresses is
assumed instead of the prior constant vertical stresses.
3.6.1 Derivation of Plastic Moment under Positive Flexure
One can generate the stress diagrams shown in figure 3-11 for a fully plastified HBE
cross-section of elasto-perfectly plastic steel subjected to axial compression, shear,
vertical stresses due to unequal top and bottom tension fields, and positive flexure.
-f y
hw
bf
tw
FIGURE 3-11 Stress Diagrams of Intermediate HBE Cross-Section under Positive Flexure, Axial Compression, Shear Force, and Vertical Stresses
due to Unequal Top and Bottom Tension Fields
As a reasonable approximation, linearly varying vertical stresses are assumed to act on
the HBE web. Mathematically, this assumption is expressed as:
( ) 11y yi yi
y y y f f h f h
σ σ σ + = ⋅ − + ⋅
(3-27)
where the vertical stresses at the top and bottom edges of the HBE web, 1yiσ + and yiσ ,
can be calculated based on the tension field information respectively as:
38
1 1
ybi yi
wt ω
σ + + = (3-28)
ybi yi
wt ω
σ = (3-29)
In accordance with (3-12), the tension and compression axial yield strengths at any
location in the web can be determined respectively as:
( ) ( ) ( ) 2 2 1 1 4 3 12 2 2
y yt w
σ σσ τ = + − −
y yc w
σ σσ τ = − − −
( ) 1
yi yit
yi yi w
y y y f f h f h
y y f h f h f
σ σσ
yi yic
yi yi w
y y y f f h f h
y y f h f h f
σ σσ
t w cy h y= − (3-34)
39
For the case when the neutral axis remains in the HBE web, the contribution of the top
and bottom flanges to axial fore is null since the resultant axial forces in each flange have
the same magnitude but opposite signs. Therefore, cross-section axial force equilibrium is
given as:
y h
t w c wy y t dy y t dy Pσ σ+ =∫ ∫ (3-35)
Substituting (3-32), (3-33) and (3-34) into (3-35), gives the following equation:
1
1 1 2
w
yi
t dy y y
y f h
w
y f h
σ
(3-36)
The above equation includes the integral of 2ax bx c dx+ +∫ , of which the solution is
2 2 2 2
2 4 ln 2 2 4 8
b ax ac bax bx c dx ax bx c ax b ax bx c C a a
+ −+ + = + + + + + + + +∫ (3-37)
However solving for the closed-form solution of cy from the resulting equation is not
possible. One can use some software packages such as Mathcad to solve for cy . Then,
knowing cy , the contribution of the web to the moment resistance can be determined as:
( ) ( ) 0 2 2
w c w
w c
− −
= − + − ∫ ∫ (3-38)
40
For the contribution of the flanges, pr flangeM − , and cross-section plastic moment reduction
factor, β , (3-24) and (3-25) still apply.
3.6.2 Derivation of Plastic Moment under Negative Flexure
Results can also be generated following the same procedure for the case of negative
flexure, i.e. for flexural moment acting in a direction opposite to what was considered in
the Section 3.6.1. The resulting stress diagrams are shown in figure 3-12 for a fully
plastified HBE cross-section of elasto-perfectly plastic steel subjected to axial
compression, shear, vertical stresses due to unequal top and bottom tension fields, and
negative flexure.
-σc
τw
hw
bf
tw
tf
+σt
FIGURE 3-12 Stress Diagrams of Intermediate HBE Cross-Section under Negative Flexure, Axial Compression, Shear Force, and Vertical
Stresses due to Unequal Top and Bottom Tension Fields
All the equations developed to locate the neutral axis in Section 3.6.1 remain valid except
that the integral limits of (3-36) need to be modified as shown below according to the
stress diagrams shown in figure 3-12.
41
1
1 1 2
yi
t dy y y
y f h
2
c
yi
y f h
σ
(3-39)
Solving for cy from (3-39), one can obtain the contribution of the web to the cross-
section plastic moment resistance as:
( ) ( ) 02 2
w c
h yw w pr web t w c wy
h hM y y t dy y y t dyσ σ− = − + − ∫ ∫ (3-40)
For the contribution of the flanges to the moment resistance, pr flangeM − , and cross-section
plastic moment reduction factor , β , (3-24) and (3-25) still apply.
3.6.3 FE Verification
To validate the approach developed in Sections 3.6.1 and 3.6.2 for the plastic moment
resistance of intermediate HBE under axial compression, shear force and vertical stresses
due to unequal top and bottom tension fields, a series of FE analyses were performed.
The FE models for positive and negative flexure cases are shown in figures 3-13 and 3-14
respectively.
42
ybi+1
s W21x73
FIGURE 3-13 FE Model of Intermediate HBE under Positive Flexure, Axial Compression, Shear Force, and Vertical Stresses due to Unequal Top and Bottom
Tension Fields
ωybi+1
s W21x73
FIGURE 3-14 FE Model of Intermediate HBE under Negative Flexure, Axial Compression, Shear Force, and Vertical Stresses due to Unequal Top and Bottom
Tension Fields
The material, element, mesh, boundary condition, and loading condition are the same as
those used in Section 3.5.2 except that a surface traction was applied on the HBE web to
achieve the transition between the unbalanced infill panel forces and satisfy vertical force
equilibrium, and opposite end rotations were applied in the negative flexure case. The
surface traction can be calculated as:
1ybi ybi
43
Note that, in reality, no such traction force is applied on the HBE web, as the unbalanced
infill panel forces are equilibrated similarly to a uniformly distributed load on a beam.
However, in modeling only a small beam segment as done above, application of the
traction keeps the segment in self-equilibrium and guarantee linear distribution of vertical
stresses in the HBE web in accordance with the assumption used for the analytical
procedure presented in Sections 3.6.1 and 3.6.2.
A series of analyses were conducted using the FE models described above to assess the
accuracy of the analytical procedure, and the effectiveness of the proposed model. In
these analyses, for the given axial compression and shear force, the vertical component of
the bottom tension field was kept constant and various magnitudes of the vertical
component of the top tension field were considered, from zero up to a value equal to that
of the bottom tension field. This range of infill panel forces acting on the top tension field
starts from an intermediate HBE section equivalent to an anchor HBE section in the
absence of the top tension field, and ends with the previously considered case of an
intermediate HBE section under equal top and bottom tension fields.
Comparisons between the results obtained from the FE analysis and those obtained by the
procedure proposed in Sections 3.6.1 and 3.6.2 for the positive and negative flexure cases
are shown in figures 3-15 and 3-16, respectively. It is shown that the plastic moment
resistance of intermediate HBE can be accurately estimated using the proposed
procedure. The cross-section plastic moment reduction factor varies from unity to the
minimum determined by (3-26) as a result of the increasing shear force, axial force and
vertical stresses.
Comparing the results shown in figures 3-15 and 3-16, it is also possible to observe that,
for the same combination of axial compression, shear and vertical stresses, the positive
plastic moment resistance is greater than the negative one. For example, according to
case (a) shown in figure 3-15, the plastic moment reduction factor is 0.97 for an
intermediate HBE under positive flexure for which 0.00xy yfτ = , 0.00wβ = ,
1.00yi yfσ = , and 1 0.20yi yfσ + = ; however, a smaller value of 0.86 is obtained for
plastic moment reduction factor in the negative flexure case in case (a) shown in figure 3-16
44
assuming the same loading combination. This can be explained on the basis that higher
vertical stresses are acting at the bottom of the beam segment, which is also in axial
compression in the negative flexure case. Recall from figure 3-6 that shows yielding
under biaxial loading conditions, per the von Mises criterion, that the compression axial
yield strength is more reduced by the presence of vertical stresses than the tension axial
yield strength. Therefore, the plastic moment is reduced more in the negative flexure case
than the positive flexure case.
45
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
β
0.00 0.20 0.40 0.60 0.80 1.00 σyi+1/fy
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
β
FIGURE 3-15 Plastic Moment Reduction Factor of Intermediate HBE Cross-Section under Positive Flexure, Axial Compression, Shear Force, and Linear Vertical Stresses: Analytical Predictions versus FE Results
46
0.70
0.75
0.80
0.85
0.90
β
0.00 0.20 0.40 0.60 0.80 1.00 σyi+1/fy
0.70
0.75
0.80
0.85
0.90
β
FIGURE 3-16 Plastic Moment Reduction Factor of Intermediate HBE Cross-Section under Negative Flexure, Axial Compression, Shear Force,
and Linear Vertical Stress: Analytical Predictions versus FE Results
47
3.6.4 Simplification of Analytical Procedures
Though the analytical procedures to estimate the plastic moment resistance of
intermediate HBE under unequal top and bottom tension fields were developed in
Sections 3.6.1 and 3.6.2 and verified by the FE results in Section 3.6.3, impediments
exists that may limit the wide acceptance of this approach in design. For example, there
are challenges in solving for cy from (3-36) and (3-39). The mathematic difficulty results
from the presence of non-uniform vertical stresses.
Aiming at the kind of simple equations derived to calculate plastic moment of the
intermediate HBE under equal top and bottom tension fields, for which a constant vertical
stress is assumed, it would be expedient to replace the linearly varying vertical stresses in
the case at hand by an equivalent constant vertical stress. However, it would not be
appropriate to use a constant vertical stress of magnitude equal to the average stress of
the linearly varying vertical stresses as an approximation, because such a unique
approximate constant vertical stress would result in identical positive and negative cross-
section plastic moments. This is inconsistent with the prior observations on plastic
moment resistances of intermediate HBEs under positive and negative flexures as shown
in figures 3-15 and 3-16, respectively.
As a compromise between simplicity and accuracy, to consider the different effects of
vertical stresses on positive and negative cross-section plastic moments, the magnitudes
of those stresses at the three-fourth and one-fourth points of the linearly varying stress
diagram, as shown in figure 3-17 (i.e. mean of the average and the minimum, and mean
of the average and the maximum, respectively), are taken as the magnitudes of the
approximate constant vertical stresses for the positive and negative flexure cases,
respectively. Mathematically, the magnitudes of these equivalent constant vertical stress
distributions can be expressed as:
( ) ( )1 1 1 1 2 4y un yi yi yi yiσ σ σ σ σ− + += + ± − (3-42)
48
"-" and "+" are employed in (3-42) for positive and negative flexure cases respectively.
Then, the procedures to determine the plastic moment of intermediate HBE under equal
top and bottom tension fields (as presented in Section 3.5.1) can be used.
σyi
FIGURE 3-17 Simplification of Vertical Stress Distribution
To check the adequacy of this model, the results obtained using the above simplified
procedure are compared with those using the more rigorous approach developed earlier
(Sections 3.6.1 and 3.6.2 for positive and negative flexure cases respectively) in figures
3-18 and 3-19, respectively. It is found that the simplified procedure provides reasonable
and relatively efficient estimates for the cross-section plastic moment reduction factor,
although slightly less accurate results are observed in the negative flexure case.
49
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
β
0.00 0.20 0.40 0.60 0.80 1.00 σyi+1/fy
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
β
FIGURE 3-18 Plastic Moment Reduction Factor of Intermediate HBE Cross-Section under Positive Flexure, Axial Compression, Shear Force,
and Linear Vertical Stresses: Analytical Predictions versus Simplified Approach
50
0.70
0.75
0.80
0.85
0.90
β
0.00 0.20 0.40 0.60 0.80 1.00 σyi+1/fy
0.70
0.75
0.80
0.85
0.90
β
FIGURE 3-19 Plastic Moment Reduction Factor of Intermediate HBE Cross-Section under Negative Flexure, Axial Compression, Shear Force,
and Linear Vertical Stresses: Analytical Predictions versus Simplified Approach
51
3.7 Additional Discussions on Anchor HBEs
The procedure developed in Section 3.6 to calculate plastic moment of intermediate HBE
can also be applied for anchor HBE, since an anchor HBE is only a special case of an
intermediate HBE for which the tension field is only applied on one side. However,
anchor HBEs deserve further treatment here since the analytical, FE, and simplified
results shown in figures 3-15, 3-16, 3-18 and 3-19 respectively, only provide results for
anchor HBE in limited scenarios (i.e. only loading scenarios corresponding to the left
ends of those curves shown in figures 3-15, 3-16, 3-18 and 3-19). Additional examples
are provided in this section to further confirm that the procedures proposed in Section 3.6
for estimating the plastic moment of intermediate HBEs remain valid for anchor HBEs.
( )
σ σ = ⋅ −
Accordingly, the analytical procedures presented in Sections 3.6.1 and 3.6.2, and
simplified approach presen

seismic behavior and design of boundary frame members of

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